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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmoeqi | Structured version Visualization version GIF version |
Description: Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
rmoeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
rmoeqi | ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmoeqi.1 | . . . . 5 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2829 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | 2 | anbi1i 623 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)) |
4 | 3 | mobii 2544 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) |
5 | df-rmo 3376 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-rmo 3376 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
7 | 4, 5, 6 | 3bitr4i 303 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ∃*wmo 2534 ∃*wrmo 3375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-mo 2536 df-cleq 2725 df-clel 2812 df-rmo 3376 |
This theorem is referenced by: disjeq1i 36134 |
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